Aspects of Singularities
University Lille 1
May 30 - June 1, 2012

Program                            Conference room:  "Salle des Réunions", M2 building

Wednesday, Mai 30		Thursday, Mai 31		Friday, June 1
-	-	9h10-10h00	Denkowski	9h00-9h50	Pe Pereira
-	-	*	coffee break	10h00-10h50	Borodzik
-	-	10h25-11h15	Kurdyka	*	coffee break
-	-	11h30-12h20	Tanabe	11h15-12h15	Colloquium talk   A'Campo
13h30	coffee	***	lunch break	***	lunch break
14h00-14:50	Cluckers	14h15-15:05	Steenbrink	14h15-15:05	Pichon
*	coffee break	*	coffee break	*	coffee break
15h20-16h10	Veys	15:30-16h20	Teissier	15:30-16h20	Parameswaran
16h25-17h15	Mourtada	16h30-17h20	Loeser	16h30-17h20	Siersma
17h30-18:20	Faber	17h30-18h20	Le Quy Thuong	17h30-18h20
**	pot	***	cocktail

List of registered participants

Norbert A'Campo (Basel) Karim Bekka (Rennes) Ana Belen de Felipe (Paris) Maciej Borodzik (Warsaw) Arnaud Bodin (Lille) Ying Chen (Lille) Raf Cluckers (Lille) Georges Comte (Chambery) Maciej Denkowski (Cracovie) Nicolas Dutertre (Marseille) Abdelghani El Mazouni (Lens)

Eleonore Faber (Wien) Goulwen Fichou (Rennes) Pedro D. Gonzalez Perez (Madrid) Youssef Hantout (Lille) Krzysztof Kurdyka (Chambery) Monique Lejeune-Jalabert (Versailles) Ann Lemahieu (Lille) François Loeser (Paris) Mohammad Moghaddam (Teheran) Hussein Mourtada (Paris) Le Quy Thuong (Paris) Parameswaran A.J. (TIFR, Mumbai)

Maria Pe Pereira (Paris) Anne Pichon (Marseille) Camille Plénat (Marseille) Patrick Popescu-Pampu (Lille) Michel Raibaut (Paris) Dirk Siersma (Utrecht) Joseph Steenbrink (Nijmegen) Susumu Tanabé (Istanbul) Bernard Teissier (Paris) Mihai Tibar (Lille) David Trotman (Marseille) Wim Veys (Leuven)

 

Conferences

Norbert A'Campo  (University  Basel)           Colloquium talk :  Nouveaux outils topologiques et géométriques dans l'étude des singularités

Maciej Borodzik  (University of  Warsaw)                    Topological aspects of spectra of singularities
Abstract. 
We shall present a relationship between topological invariants links of hypersurface singularities and spectra of the corresponding singular points. In case of plane curve singularities, we can fully recover the spectrum from the classical link invariants. Furthermore, we show that classical skein relations in the link theory gives rise, via suitably applied Morse theory, to the proof of various semicontinuity properties of spectra. We shall finish the talk by pointing out some higher dimensional generalizations. This is a joint project with A. Nemethi.

Susumu Tanabe (Galatasaray University)      Period integrals for complete intersection varieties
Abstract.
In this talk, we will discuss about concrete expression of solutions to Gauss-Manin system or Picard-Fuchs equation associated to affine complete intersection varieties. Our main interest will be focused on several cases where the concrete monodromy representation of the solutions is available. As an example of our investigations on Horn type hypergeometric functions, we show the following. Let $Y$ be a Calabi-Yau complete intersection in a weighted projective space. The space of quadratic invariants of the (reduced) hypergeometric group associated with the period integrals of the mirror CI variety X to Y is one-dimensional and spanned by the Gram matrix of a split-generator of the derived category of coherent sheaves on Y with respect to the Euler form.

Anne Pichon  (Université de la Méditérranée)    The bilipschitz geometry of a normal complex surface
Abstract.
This is a joint work with Lev Birbrair and Walter Neumann. We study the geometry of a normal complex surface $X$ in a neighbourhood of a singular point $p \in X$. It is well known that  for all sufficiently small $\epsilon>0$ the intersection of $X$ with the sphere $S^{2n-1}_\epsilon$ of radius $\epsilon$ about $p$ is transverse, and $X$ is therefore locally "topologically conical'' i.e., homeomorphic to the cone on its link $X\cap S^{2n-1}_\epsilon$. However,  as shown by Birbrair and Fernandez, $(X,p)$  need not be "metrically conical'', i.e. bilipschitz equivalent to a standard metric cone  when $X$ is equipped with the Riemanian metric induced by the ambient space.   In fact,  it was shown by Birbrair, Fernandez and Neumann that it rather rarely is.
I will present,  a complete classification of the bilipschitz geometry of $(X,p)$. It starts with a decomposition of a normal complex surface singularity into its "thick'' and `"thin'' parts. The former is essentially metrically conical, while the latter shrinks  rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical. Then the complete classification consists of a refinement of the thin part into geometric pieces. I will describe it on an example, and I will present a list of open problem related with this new point of view on classifying complex singularities.

Eleonore Faber (Universität Wien)               Splayed divisors: transversality of singular hypersurfaces
Abstract
  In this talk we present a natural generalization of transversally intersecting smooth hypersurfaces in a complex manifold: hypersurfaces, whose components intersect in a transversal way but may be themselves singular. We call these hypersurfaces "splayed" divisors. A splayed divisor is characterized by a property of its Jacobian ideal. Another characterization is in terms of K.Saito's logarithmic derivations.  As an application we consider the question of characterizing a normal crossing divisor by its Jacobian ideal.

Hussein Mourtada  (Université Paris Diderot)      Jet schemes of normal toric surfaces
Abstract 
For an integer m >0 we will determine the irreducible components of the m-th jet scheme of a normal toric surface S. We give formulas for the number of these components and their dimensions. When m varies, these components give rise to projective systems, to which we associate a weighted oriented graph. We prove that the data of this graph is equivalent to the data of the analytical type of S. Besides, we classify these irreducible components by an integer invariant that we call index of speciality. We prove that for m large enough, the set of components with index of speciality 1 is in one-to-one correspondance with the set of exceptional divisors that appear on the minimal resolution of S.

François Loeser   (Université Pierre et Marie Curie)      Fixed points of iterates of the monodromy
Abstract 
With Jan Denef we proved a formula relating arc spaces to fixed points of iterates of the monodromy. The proof was by explicir computation on a resolution. We shall present a recent work with Ehud Hrushovski that provides a new - geometric - proof using Lefschetz fixed point formula and non-archimedean geometry. If times allow we shall end by discussing how similar methods provide a new construction of the motivic Milnor fiber.

Joseph Steenbrink  (Radboud Universiteit Nijmegen)                        Function germs on toric singularities
Abstract
We study function germs on toric varieties which are nondegenerate for their Newton diagram. We express their motivic Milnor fibre in terms of their Newton diagram. We establish an isomorphism between the cohomology of the Milnor fibre and a certain module constructed from differential forms. This module is equipped with its Newton filtration. We conjecture that its Poincare polynomial is equal to the spectrum of the function germ. We will illustrate this by several examples.

Le Quy Thuong  (Université Pierre et Marie Curie)    Some approaches to the Kontsevich-Soibelman integral identity conjecture

Abstract

It is well known that the integral identity conjecture is one of the key foundations in Kontsevich-Soibelman's theory of motivic Donaldson-Thomas invariants for 3-dimensional Calabi-Yau varieties. In this talk, we shall mention some approaches to the conjecture in the algebro-geometric and arithmetic-geometric points of view. Namely, we consider the regular, adic and formal versions and give some proofs under certain conditions.

Raf Cluckers  (Université Lille 1)     An overview on recent progress related to (sub-)analytic functions
Abstract
We will give a tour around recent developments about (sub-)analytic functions, both in a real setting and in a non-archimedean setting. The topics will include aspects of integration, parameterizations, Lipschitz continuity, and a link with number theory in analogy to Pila's work with e.g. Bombieri in which real topological techniques are used to bound the number of rational points on, say, the graph of an analytic function.

Wim Veys   (Katholieke Universiteit Leuven)    Bounds for log-canonical thresholds and exponential sums
Abstract
This is joint work with Raf Cluckers. For any polynomial f with rational coefficients, we study the bounds conjectured by Denef and Sperber for local exponential sums, modulo m-th powers of a prime, associated to f. We show that such bounds hold unconditionally for several small values of m, with the log-canonical threshold of f in the exponent. Key ingredients are some new bounds for log-canonical thresholds.

Maria Pe Pereira       Nash problem for surfaces
Abstract   
Let π : (X, E) → (X, O) be a resolution of singularities of a singularity (X, O). Take the decomposition of the exceptional divisor E =\cup_i Ei.  Given any arc γ : (C, 0) → (X, SingX) one can consider the lifting γ : (C, 0) → (X, E).  Nash considered the set of arcs whose lifting γ meets a fix divisor Ei , that is γ (0) ∈ Ei , and proved that its closure is an irreducible set of the space of arcs. Nash’s question is whether for the essential divisors Ei these subsets of arcs are in fact irreducible components of the space of arcs or not. He conjectured that the answer was yes for the case of surfaces and suggested the study in higher dimensions. Recently we solved the conjecture for the surface case in a joint work with J. Fernandez de Bobadilla. I will give an introduction to the problem and details of a the proof for the normal surface case.

Maciej Denkowski  (Jagellonian University Cracow)                On the exceptional (central) set
Abstract
For a given subanalytic (or definable in some o-minimal structure) closed set $M\subset\mathbb{R}^n$ we are interested in the multifunction assigning to each point $x\in\mathbb{R}^n$ the compact set $m(x)\subset M$ consisting of the points $y\in M$ realizing the Euclidean distance $\mathrm{dist}(x,M)$, as well as in the structure of the exceptional set $E$ of points for which $\#m(x)>1$, i.e. there is more than one closest point to $x$. The exceptional set $E$ conveys interesting information about the singularities of $M$. This study is closely related to such notions as skeletons, central sets and conflict sets. Part of it is a joint project with Lev Birbrair.

Krzysztof Kurdyka   (Université de Savoie)      Reaching  generalized critical values of a polynomial
Abstract
We give an algorithm to compute the set of asymptotic (respectively generalized)  critical value of a polynomial both in the complex and real case. The algorithm uses a finite dimensional space of  rational arcs along which we can reach  all asymptotic  (respectively  generalized)  critical values. Joint work with Z. Jelonek.

Dirk Siersma    (Universiteit Utrecht)        Projective Hypersurfaces with non-isolated singularities
Abstract
We intend to discuss the influence of singularities on the homology groups of projective hypersufaces. We use the vanishing homology of the singular set in order to compare with the general (smooth) projective hyper surface. First we discuss what is known about isolated singularities. Next we investigate the case of a one dimensional singular set, derive a formula for the Euler characteristic and some information about the homology groups. We treat some examples in detail. This is joint work (in progress) with Mihai Tibar.

Parameswaran A.J.  (TIFR Mumbai)    On the geometry of regular maps from a quasi-projective surface to a curve
Abstract
In case of a polynomial function $P : \bC^2 \to \bC$, Miyanishi and Sugie proved that the global monodromy group acting on $H_1(F)$ is trivial if and only if the general fibre of $P$ is rational and $P$ is "simple".  Dimca showed in the following that in this statement the monodromy group can be replaced by the monodromy at infinity (i.e. around a very large cercle in $\bC$).  We explore here some consequences of the triviality of the monodromy group, in Hodge theoretic terms. Joint with M. Tibar.